Let it next be required to find the sum of the numbers 29 and 68:

tens

units
9

2

6

8

This result, 8 tens and 17 units, rightly represents the sum of the units and tens contained in the given number; but it appears in a form requiring some modification to bring it into accordance with the decimal scale of notation; for it is a principle of this scale that ten units of any order make one unit of the order next higher; and here are seventeen units. The change to be made is this :

Since 17 units=1 ten+7 units.

The number 8 tens +17 units=8 tens+1 ten+7 units. That is, it equals 9 tens+7 units=97.

* .. 29+68=97.

Explanation. Having written the numbers under each other, in such manner that the units of the same order may occupy the same vertical column, the addition is made thus:

1st, of the units: 8+4=12. 12+6=18. 18+7=25 units. 25 units=2 tens and 5 units. The 5 units are recorded under the column of units; and the two tens carried forward to the column of tens.

2d, of the tens: 2+9=11. 11+9=20. 20+8=28. 28+6=34 tens. 34 tens 3 hundreds and 4 tens. The 4 tens are written under the column of tens, and the 3 hundreds carried forward to the column of hundreds.

3rd, of the hundreds: 3+2=5. 5+3=8. 8+1=9 hundreds. This 9 being written in its proper place, the operation is completed, and the sum of the given numbers found to be 945.

21. Having found the sum of any column, and written the units contained in that sum in their proper place, the tens, reserved for combination with the next column, may be conveniently recorded by means of a small figure written under the units of that particular order from which they are carried. An advantage of this practice is, that if, after the sums of several columns have been found, any interruption occur, and the sum of the column under process of addition be lost, the number carried from the preceding column is not lost along with it.

Without some such expedient it might, in cases like the one imagined, be necessary to resume the work from the beginning.

In the last example the 2 written under the five units, and the 3 under the 4 tens, are instances of the precaution recommended.

The work and illustrations of the last example appear sufficient to direct the mode of procedure in the addition of any whole numbers whatever. The words units, tens, hundreds, placed in the preceding examples over the different orders of units, were so placed to give distinctness to the explanations. The learner, after fully understanding the rationale of addition, may omit them from his work.

22. In finding the sum of several numbers, each composed of two or more orders of units, the addition is always commenced with the extreme column on the right, that is, with the simple units.

*The symbol.. is put for the word therefore.

The reason for so beginning is, that the tens, which may arise in the addition of the units, can thus be at once taken into account in adding the column of tens; the hundreds arising from the column of tens, in that of the hundreds, &c.

If the extreme left column be taken first, and the addition made from left to right, either new additions must be made to combine the tens arising from the units of any order with the units of the next higher; or the result must remain with more than nine units in, at least, some of the orders.

But for these circumstances, it would be matter of indifference whether, in adding numbers together, the process were performed from right to left or from left to right.

For the sake of illustration, the addition in the following example is made from left to right:

1st. 7+3=10, 10+8=18, 18+5=23 thousands.
2d. 6+5=11, 11+7=18, 18+3=21 hundreds.
3d. 8+1=9, 9+9=18, 18+4=22 tens.
4th. 4+2=6, 6+0=6, 6+6=12 units.

The sum is therefore made up of 23 thousands

= 23000 21 hundreds = 2100

To reduce the result to a single number, a second addition is necessary. As the sum of no column, in the present instance, exceeds nine, this second addition, whatever way begun, completes the work. But, supposing always the addition to be made from left to right, cases may occur in which three or more operations will be required to reduce tens of an inferior order into units of the next higher; that is, to reduce the sums of the different columns into one number.

The preceding remarks and demonstrations are, it is presumed, sufficient to render intelligible the principles and operations of addition.

A General Rule and Examples are annexed.

23. General Rule for the addition of whole numbers.

Write the numbers to be added together under each other in successive lines, so arranged that all the figures representing units of the same order may occupy the same vertical column.

Draw a line under the last number, to separate the addenda from the result. Then, beginning with the right column or units of the first order, find the sum of the figures composing this column: write the units of the sum under the column added, and reserve the tens, if any, for combination, as units of the second order, with the second column, or units of the second order. Add this column, in like manner place the units of the sum below it, and reserve the tens for combination with the third column. In the same manner proceed with a third, a fourth.... to the extreme column on the left. The result of this operation is the sum of the given numbers.

24. Examples in the addition of whole numbers. Required the sums of the following numbers:

1st. 5943, 7685, 2768, and 9374?.

2d. 38796, 67438, 54874, and 89657 ?......
3d. 73564, 89753, 46586, 95845, and 38479?.
4th. 4789, 6543, 2101, and 9876?....

......Ans. 25770.

..Ans. 250765.

.......Ans. 344227. ..Ans. 23309.

5th. 123456, 789012, 345678, 901234, and 567890?...Ans. 2727270. 6th. 2574, 83965, 1781, 479, and 8698?........

.Ans. 97497.

7th. 15893, 7586, 59948, 479, 3548, and 95 ?.............. Ans. 87549. 8th. 456, 7890, 12345, 67, 890123, and 4?.............. Ans. 910885. 9th. 49876534, 15798249, 6789012, and 4789656?...Ans. 77253451. 10th. 7690458, 7690458, 7690458, and 7690458?...Ans. 30761832. 11th. 97, 764, 386, 17578, 8997, and 59604?.. ...Ans. 87426. 12th. 8759076, 605895, 961704, 87289, 743998, 6657, and 7985065?... ..Ans. 19149684.

13th. 8796, 5348, 284, 7963, 57, 9, 154, 9598, 358, and 85?

Ans. 32652.

14th. 574, 896, 78, 985, 347, 7584, 59, 870, 5794, 648, 1842, 31, 8628, and 779?......... Ans. 29115.

15th. 689, 748, 986, 577, 854, 98, 379, 956, 28, 989, 497, 588, 867, 179, 996, 755, and 778 ?....... ........Ans. 10964.

25. The addition of two numbers is indicated by interposing the symbol +between them. Thus 9+7 is an expression signifying that the numbers 9 and 7 are to be combined into one number.

When general, instead of particular, expressions of number are employed (Art. 11) the same notation is used.

If a represent one number, and b another, the sum of a and b is written a+b.

In like manner, the sum of a, b, c, is denoted by the expression a+b+c. If any quantity, a, is added to itself, the sum of the repetitions may be expressed in the same manner; as, a+a; a+a+a.

But it is to be observed that a signifies once the number à;

that once a,+ once a, = twice a=2a:

that once a,+ once a, + once a, = three times a=3a, &c.

Hence the practice is, not to write the number as many times as it is repeated, but once only; and to express the number of repetitions by a figure prefixed to the a, or other general expression of number.

Numbers which (like 2 and 3 in the present instances) express the number of repetitions of a quantity, are termed co-efficients of that quantity. Again, the sum of the quantities 2a and 3a may be indicated by the expression 2a+3a.

But since 2a=a+a

and 3a=a+a+a.

Therefore 2a+3a=a+a; +a+a+a.

or 2a+3a=5a.

2a, 3a, are expressions which signify that the quantity a is to be taken in the one instance two, and in the other three, times; that is, as often as the co-efficients contain simple units.

If the numeral co-efficients 2, 3, are replaced by the general expressions m, n (termed literal co-efficients), then, ma or m times a, indicates that the number a is to be repeated as often as the co-efficient m contains simple units; and na that the number a is again to be repeated as often as the co-efficient n contains simple units.

Whence, if a repeated m times is increased by a repeated n times, the sum is equal to a repeated as often as m and n together contain simple units.

But the number of simple units contained in m and n is indicated by the expression m+n. Consequently ma+na=(m+n) a.

To signify that a is repeated m+n times, and not m times only, or n times only, the sum of the co-efficients m, n, is written thus (m+n).

26. Quantities expressed by the same letters, or by the same combinations of the same letters, are termed like or similar: quantities expressed by different letters, or by different combinations of the same letters, unlike or dissimilar. From this definition, and the conclusions of Article 25, it follows that the addition of unlike quantities is indicated by connecting the quantities together with the symbol +, and the addition of like quantities, by adding the co-efficients together, and writing their sum before the common quantity. 26'. Each of the quantities composing an expression such as a+b+c, or a+b-c, is called a term.

A quantity composed of one term only is called a monomial; a quantity composed of two terms, a binomial; a quantity composed of three terms, sometimes a trinomial; but generally a quantity composed of more terms than two is called a polynomial.

Exercises in the addition of general expressions of quantity. 1st. Required the sum of 3a, 5a, a, 6a, 17a ?.......

2d.

3d.

4th.

5th.

6th.

7th.

8th.

9th.

10th.

....Ans. 32a. ma, 2ma, 5na, and a ?....................... Ans. (3m+5n+1)a. a, 5a, ma ?....

a, mb, 6d, 3a, b, and 4g?

.................Ans. (6+m)a.

Ans. 4a+6d+(m+1)b+4g.

a+b, ma+nc, 3b+5c?

a+2ab, and ab+b?

Ans. (1+m)a+4b+(n+5)c.

Ans. a+2ab+b, or a+(2a+1)b.

a+2ab+db, and ab+2db+b?

Ans. a+3ab+3db+b, or a+(3a+3d+1)b, or a+ {3(a+d)+1}b. a+3b+5, b+5a+17, and 2a+10?

Ans. 8a+4b+32.

27. Two unequal collections of units are given, and it is required to find how many more are contained in the greater collection than in the less: or, how many units of the greater collection are left when as many as are contained in the less have been taken from it. Such is the problem of subtraction.

**

As a particular case, let it, for the sake of illustration, be supposed that a boy who had twelve apples gave five away; the question is, How many had he left?

twelve less one, or eleven,
eleven less one, or ten,
ten less one, or nine,
nine less one, or eight,
eight less one, or seven.

Whence, having from twelve apples given away five, the boy had seven

left.

A problem is a question which requires a solution.

In this example the subtraction is effected by taking from the whole number of apples, first one, then a second, a third, a fourth, a fifth. Also, the steps of the operation are traced and the final result enunciated by means of a series of names of numbers, commencing with that of the greater quantity, and descending by successive subtractions of unity until the name corresponding to the sum of the units taken away represents the less quantity. The name arrived at by these successive decompositions is that of the result sought.

28. This result is termed Remainder.

If two numbers are mentioned as being unequal, but without distinction of the greater, the term Difference is given to that number which constitutes the inequality. But if the greater be distinguished, instead of difference, the proper term is Excess.

29. Subtraction, like addition, is manifestly an application of the principles of numeration. It requires that the names of numbers be known in a descending series; or in the order contrary to that of their formation. Addition being performed by means of the ascending series of numbers, the one operation is the reverse of the other.

The minuend and subtrahend may be abstract or concrete numbers: if concrete, the numbers must express collections of individuals of the same kind; the operation, being conducted through the intervention of names, is the same for concrete as for abstract numbers.

30. The difference of small numbers is found by decomposing the minuend into two numbers, of which one is the subtrahend and the other the remainder; and this decomposition is effected by taking from the minuend successive units, until the number taken away is equal to the subtrahend.

But as, in finding the difference of large numbers, this method, by the continued subtraction of simple units, would be impracticable, the conventions of numeration which were employed to facilitate addition are in like manner rendered subsidiary to the operation of subtraction. Thus both the quantities are considered to be decomposed into collections of simple units, tens, hundreds, &c.; the units of the subtrahend are taken from the units of the minuend, the tens from the tens . . . . . . in fine, the units of every order contained in the former from the units of the correspondent order in the latter. The result of these partial subtractions is the remainder sought.

31. That a remainder, thus obtained, correctly expresses the difference of the given numbers, is proved by the considerations following: :

The whole being equal to the sum of all its parts, the minuend and subtrahend are respectively equal to the sum of the units, tens, &c. into which they are decomposed.

Áll the units, tens, &c. of the subtrahend are taken from the units, tens, &c. of the minuend, each from each; that is, all the parts of the subtrahend are taken from the minuend. But to take away all the component parts of any quantity, amounts to the subtraction of the whole of that quantity. Whence the whole of the subtrahend is taken from the minuend, and consequently the remainder obtained by taking the units of the former from those of the latter, the tens from the tens, &c., is, in effect, the correct expression of that number of units by which the greater quantity exceeds the less.

32. Reviewing the whole subject, it appears that the subtraction of one unit from another, or of one small number from another, is accomplished by the process of decomposition explained in Article 27; and that the subtraction of large numbers (which has been reduced to that of the successive orders of units in the subtrahend from the corresponding orders in the minuend) is, in effect, performed by repetitions of the same elementary process.

33. The subtraction of one small number from another may be, at first, effected by means of sensible objects, as was recommended in addition; or, they who prefer it, may use the following